Proceedings of the 44th Symposium on Ring Theory and ...

More documents

Recommendations

Info

References [1] M. Hoshino, Y. Kato, J. Miyachi, On t-structure and torsion <strong>the</strong>ories induced by compact objects, J. Pure and App. Algebra 167 (2002), 15–35. Institute <strong>of</strong> Ma<strong>the</strong>matics University <strong>of</strong> Tsukuba Ibaraki 305-8571 JAPAN E-mail address: abeh@math.tsukuba.ac.jp –5–
DIMENSIONS OF DERIVED CATEGORIES TAKUMA AIHARA AND RYO TAKAHASHI Abstract. Several years ago, Bondal, Rouquier and Van den Bergh introduced <strong>the</strong> notion <strong>of</strong> <strong>the</strong> dimension <strong>of</strong> a triangulated category, and Rouquier proved that <strong>the</strong> bounded derived category <strong>of</strong> coherent sheaves on a separated scheme <strong>of</strong> finite type over a perfect field has finite dimension. In this paper, we study <strong>the</strong> dimension <strong>of</strong> <strong>the</strong> bounded derived category <strong>of</strong> finitely generated modules over a commutative Noe<strong>the</strong>rian ring. The main result <strong>of</strong> this paper asserts that it is finite over a complete local ring containing a field with perfect residue field. 1. Introduction The notion <strong>of</strong> <strong>the</strong> dimension <strong>of</strong> a triangulated category has been introduced by Bondal, Rouquier and Van den Bergh [4, 14]. Roughly speaking, it measures how quickly <strong>the</strong> category can be built from a single object. The dimensions <strong>of</strong> <strong>the</strong> bounded derived category <strong>of</strong> finitely generated modules over a Noe<strong>the</strong>rian ring and that <strong>of</strong> coherent sheaves on a Noe<strong>the</strong>rian scheme are called <strong>the</strong> derived dimensions <strong>of</strong> <strong>the</strong> ring and <strong>the</strong> scheme, while <strong>the</strong> dimension <strong>of</strong> <strong>the</strong> singularity category (in <strong>the</strong> sense <strong>of</strong> Orlov [12]; <strong>the</strong> same as <strong>the</strong> stable derived category in <strong>the</strong> sense <strong>of</strong> Buchweitz [5]) is called <strong>the</strong> stable dimension. These dimensions have been in <strong>the</strong> spotlight in <strong>the</strong> studies <strong>of</strong> <strong>the</strong> dimensions <strong>of</strong> triangulated categories. The importance <strong>of</strong> <strong>the</strong> notion <strong>of</strong> derived dimension was first recognized by Bondal and Van den Bergh [4] in relation to representability <strong>of</strong> functors. They proved that smooth proper commutative/non-commutative varieties have finite derived dimension, which yields that every contravariant cohomological functor <strong>of</strong> finite type to vector spaces is representable. As to upper bounds, <strong>the</strong> derived dimension <strong>of</strong> a ring is at most its Loewy length [14]. In particular, Artinian rings have finite derived dimension. Christensen, Krause and Kussin [6, 9] showed that <strong>the</strong> derived dimension is bounded above by <strong>the</strong> global dimension, whence rings <strong>of</strong> finite global dimension are <strong>of</strong> finite derived dimension. In relation to a conjecture <strong>of</strong> Orlov [13], a series <strong>of</strong> studies by Ballard, Favero and Katzarkov [1, 2, 3] gave in several cases upper bounds for derived and stable dimensions <strong>of</strong> schemes. For instance, <strong>the</strong>y obtained an upper bound <strong>of</strong> <strong>the</strong> stable dimension <strong>of</strong> an isolated hypersurface singularity by using <strong>the</strong> Loewy length <strong>of</strong> <strong>the</strong> Tjurina algebra. On <strong>the</strong> o<strong>the</strong>r hand, <strong>the</strong>re are a lot <strong>of</strong> triangulated categories having infinite dimension. The dimension <strong>of</strong> <strong>the</strong> derived category <strong>of</strong> perfect complexes over a ring (respectively, a quasi-projective scheme) is infinite unless 2010 Ma<strong>the</strong>matics Subject Classification. Primary 13D09; Secondary 14F05, 16E35, 18E30. Key words and phrases. dimension <strong>of</strong> a triangulated category, derived category, singularity category, stable category <strong>of</strong> Cohen-Macaulay modules. The second author was partially supported by JSPS Grant-in-Aid for Young Scientists (B) 22740008. –6–
Page 1: Proceedings <stron
Page 5 and 6: Organizing Committee of</st
Page 7 and 8: Polycyclic codes and sequential cod
Page 9 and 10: Preface The 44th <
Page 11 and 12: 8:40-9:30 Dan ZachariaSyracuse Univ
Page 13 and 14: The 44th S
Page 15 and 16: Tuesday September 27 8:40-9:30 Atsu
Page 17 and 18: Conversely, let (T , F) be a stable
Page 19: Then T = gen(X) and X is Ext-projec
Page 23 and 24: Notation 4. (1) Let A be an abelian
Page 25 and 26: of finitely genera
Page 27 and 28: is the map given b
Page 29 and 30: (2) Let R be a commutative ring whi
Page 31 and 32: QUIVER PRESENTATIONS OF GROTHENDIEC
Page 33 and 34: Proposition 3. Let C be a category,
Page 35 and 36: Example 11. Let Q be the</s
Page 37 and 38: and Gr(X ′ ) is given by
Page 39 and 40: particular, the de
Page 41 and 42: Proposition 1 and 2 leave us with d
Page 43 and 44: 4. Examples Example 6. Let M = M(A
Page 45 and 46: EXAMPLE OF CATEGORIFICATION OF A CL
Page 47 and 48: The aim of <strong
Page 49 and 50: X ∈ mod Π Q S 1 S 2 S 3 1 ❁
Page 51 and 52: The previous proposition has a dual
Page 53 and 54: Computing inductively all t
Page 55 and 56: Example 23. We have ⎛ ⎞ ⎛ ⎞
Page 57 and 58: classification of
Page 59 and 60: quiver Q over K, and let D b (H) <s
Page 61 and 62: Then Q ∈ Q 17 . Suppose char K
Page 63 and 64: Moreover the Hochs
Page 65 and 66: DERIVED AUTOEQUIVALENCES AND BRAID
Page 67 and 68: 3. Periodic Twists We now describe
Page 69 and 70: We now specialise to the</s
Page 71 and 72:
and τ − n := Ext n Λ(DΛ, −)
Page 73 and 74:
where r v ij := a i a j −a j a i
Page 75 and 76:
ON A DEGENERATION PROBLEM FOR COHEN
Page 77 and 78:
We make several othe</stron
Page 79 and 80:
Let A be a commutative Gorenstein r
Page 81 and 82:
3. Extended orders In the</
Page 83 and 84:
WEAK GORENSTEIN DIMENSION FOR MODUL
Page 85 and 86:
1.2. Introduction. The notion <stro
Page 87 and 88:
e 2 A z 1 −→ X → 0 in mod-A,
Page 89 and 90:
5. Gorenstein algebras In this sect
Page 91 and 92:
ON Ω-PERFECT MODULES AND SEQUENCES
Page 93 and 94:
when R is a Nakayama algebra <stron
Page 95 and 96:
says that components of</st
Page 97 and 98:
then we obtain a c
Page 99 and 100:
Proposition 16. Let C s be a stable
Page 101 and 102:
3.1. ZD ∞ -components. We assume
Page 103 and 104:
Theorem 26. Let R be a local artini
Page 105 and 106:
where each f i is an irreducible ep
Page 107 and 108:
QUANTUM UNIPOTENT SUBGROUP AND DUAL
Page 109 and 110:
{F i } i∈I and q-Serre relations
Page 111 and 112:
Theorem 8. (1)Let U − q (w) → U
Page 113 and 114:
E-mail address: ykimura@kurims.kyot
Page 115 and 116:
Indeed, (1, 2, 3, 4) {1,2} −−
Page 117 and 118:
By comparing this the</stro
Page 119 and 120:
(a) (b) Proposition 17. Let G be a
Page 121 and 122:
POLYCYCLIC CODES AND SEQUENTIAL COD
Page 123 and 124:
⎛ ⎞ 0 0 c 0 1 c t D c = ⎜ 1
Page 125 and 126:
References [1] D. Boucher and P. So
Page 127 and 128:
2. Dimension of tr
Page 129 and 130:
APR TILTING MODULES AND QUIVER MUTA
Page 131 and 132:
(1) Q ′ is a quiver obtained from
Page 133 and 134:
1 arrows of ˜µ L
Page 135 and 136:
[11] S. Fomin, A. Zelevinsky, Clust
Page 137 and 138:
Theorem. Assume r is a common prime
Page 139 and 140:
References [1] Apostol, T. M., The
Page 141 and 142:
e(n, s) n = 6 7 8 s = 1 36 63 120 2
Page 143 and 144:
Now we will show that the</
Page 145 and 146:
is equal to ( n 2s−1 ); hence we
Page 147 and 148:
THE NOETHERIAN PROPERTIES OF THE RI
Page 149 and 150:
Proposition 4 (Paper III, Propositi
Page 151 and 152:
HOCHSCHILD COHOMOLOGY OF QUIVER ALG
Page 153 and 154:
(4) In [10], Green, Snashall and So
Page 155 and 156:
4. Quiver algebras defined by two c
Page 157 and 158:
[6] L. Evens, The cohomology ring <
Page 159 and 160:
Then there is an i
Page 161 and 162:
• ( ˜F , ˜m) = ({ (1, 3), (2, 3
Page 163 and 164:
Corollary 7 ([16, Theorem 3.4]). Th
Page 165 and 166:
We have the direct
Page 167 and 168:
We say a CW complex is regular, if
Page 169 and 170:
SHARP BOUNDS FOR HILBERT COEFFICIEN
Page 171 and 172:
Let us briefly note how this paper
Page 173 and 174:
whence the set Λ
Page 175 and 176:
Proof of</
Page 177 and 178:
We have Q·H j m(A/(a 1 , a 2 , ·
Page 179 and 180:
Let B = A/U(a d−1 ). We t
Page 181 and 182:
• quiver Γ = (I, Ω) I Ω
Page 183 and 184:
B = (B τ ) relations ρ 1 , · ·
Page 185 and 186:
Definition 1. double quiver ˜Γ p
Page 187 and 188:
◦ side Γ = (I, Ω) cycle (ac
Page 189 and 190:
Definition 6. n × n A(Γ Dyn ) =
Page 191 and 192:
(2) P Lie theory
Page 193 and 194:
Example 15. Γ loop quiver λ ∈
Page 195 and 196:
P (Γ)-module i-th radical B →
Page 197 and 198:
B ∈ Λ(d) ε ∗ i (B) := dim C
Page 199 and 200:
I-graded vector space V (d) (B τ )
Page 201 and 202:
A ∗(i) l (a) = ∑n+1 t=l+1 (a i,
Page 203 and 204:
wt(a) = (−d 1 , −d 2 , −d 3 )
Page 205 and 206:
Ω Q Ω ( B Ω ; wt, ε i , ϕ
Page 207 and 208:
“Λ(d) G(d)-” Definition 25.
Page 209 and 210:
(( ( ) ( ) 0 s 0 0 0 , , , s) ( u 0
Page 211 and 212:
MATRIX FACTORIZATIONS, ORBIFOLD CUR
Page 213 and 214:
( ∂f Jac(f t t ) := C[x 1 , . . .
Page 215 and 216:
Definition 14. CM L f (R f ) Ob(C
Page 217 and 218:
4. Dolgachev Proposition 24 ([1]
Page 219 and 220:
✤ ✤ ✤ ✤ ✤ ✤ γ 1 +γ 2
Page 221 and 222:
(f, G f ) Dolgachev A Gf f Berg
Page 223 and 224:
ON A GENERALIZATION OF COSTABLE TOR
Page 225 and 226:
(2) P/t(P ) is σ-projective for an
Page 227 and 228:
(3) P is a maximal σ-coessential e
Page 229 and 230:
(2)→(1): Let σ be epi-preserving
Page 231 and 232:
GRADED FROBENIUS ALGEBRAS AND QUANT
Page 233 and 234:
Definition 2. A graded algebra A is
Page 235 and 236:
In this case, ν ∈ Aut k E induce
Page 237 and 238:
References [1] X. W. Chen, Graded s
Page 239 and 240:
The topics covered are ring-<strong
Page 241 and 242:
(2.3) soc( R R) ∼ = k⊕ s i T i
Page 243 and 244:
principal ideal Rr ⊂ ker ϖ. Thus
Page 245 and 246:
By Example 3, the
Page 247 and 248:
weight wt(x) = d(x, 0) of</
Page 249 and 250:
4.1. Fourier Transform. Gleason’s
Page 251 and 252:
5. The Extension Problem In this se
Page 253 and 254:
Because R is assumed to be Frobeniu
Page 255 and 256:
is injective for every finite left
Page 257 and 258:
linear code defined by η;
Page 259 and 260:
ψ : ε(A) → Â. In this way, C
Page 261 and 262:
REALIZING STABLE CATEGORIES AS DERI
Page 263 and 264:
2.1. Positively graded self-injecti
Page 265 and 266:
By the above resul
Page 267 and 268:
Next we consider the</stron
Page 269 and 270:
(2) There exists a triangle-equival
Page 271 and 272:
ALGEBRAIC STRATIFICATIONS OF DERIVE
Page 273 and 274:
The leaves of <str
Page 275 and 276:
Step 2: Let A be a finite-dimension
Page 277 and 278:
RECOLLEMENTS GENERATED BY IDEMPOTEN
Page 279 and 280:
(a) the adjoint tr
Page 281 and 282:
Cluster-tilting the</strong
Page 283 and 284:
INTRODUCTION TO REPRESENTATION THEO
Page 285 and 286:
is exact. Since E n ∼ = En+r , Mi
Page 287 and 288:
field of V . We de
Page 289 and 290:
(Proof) ( ) p 0 0
Page 291 and 292:
we have a sequence of</stro
Page 293 and 294:
R : CM(R⊗ k V ) → CM(R) defined
Page 295 and 296:
P = P ′ t by t is a projective R
Page 297 and 298:
SUBCATEGORIES OF EXTENSION MODULES
Page 299 and 300:
Lemma 6. Let S 1 and S 2 be Serre s
Page 301 and 302:
In the first row <
show all

Proceedings of the 44th Symposium on Ring Theory and ...

Create successful ePaper yourself

Delete template?

Save as template?